Electronic Communications in Probability

The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium

Manh Hong Duong and Julian Tugaut

Full-text: Open access

Abstract

In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 19, 10 pp.

Dates
Received: 23 August 2017
Accepted: 5 February 2018
First available in Project Euclid: 15 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1521079417

Digital Object Identifier
doi:10.1214/18-ECP116

Mathematical Reviews number (MathSciNet)
MR3779816

Zentralblatt MATH identifier
1387.60089

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 35B40: Asymptotic behavior of solutions
Secondary: 35K55: Nonlinear parabolic equations 60J60: Diffusion processes [See also 58J65] 60G10: Stationary processes

Keywords
kinetic equation Vlasov-Fokker-Planck equation free-energy asymptotic behaviour granular media equation stochastic processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Duong, Manh Hong; Tugaut, Julian. The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium. Electron. Commun. Probab. 23 (2018), paper no. 19, 10 pp. doi:10.1214/18-ECP116. https://projecteuclid.org/euclid.ecp/1521079417


Export citation

References

  • [1] A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo, P. Degond. Phase Transitions in a kinetic flocking model of Cucker-Smale type. Multiscale Model. Simul. 14, 1063–1088, 2016.
  • [2] S. Benachour, B. Roynette, and P. Vallois. Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process. Appl., 75(2):203–224, 1998.
  • [3] D. Benedetto, E. Caglioti, J. A. Carrillo, and M. Pulvirenti. A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys., 91(5–6):979–990, 1998.
  • [4] F. Bolley, A. Guillin and F. Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsitent Vlasov-Fokker-Planck equation. ESIAM: Mathematical Modelling and Numerical Analysis, 44:867–884, 2010.
  • [5] F. Bolley, I. Gentil and A. Guillin Uniform convergence to equilibrium for granular media Archive for Rational Mechanics and Analysis, 208, 2, pp. 429–445 (2013)
  • [6] L. L. Bonilla, J. A. Carrillo and J. Soler Asymptotic behavior of an initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck system SIAM J. Appl. Math., Vol. 57, No 5, pp. 1343–1372 (1997)
  • [7] F. Bouchut and J. Dolbeault On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Differential and Integral Equations, 8(3):487–514, 1995.
  • [8] J. A. Carrillo, R. J. McCann, and C. Villani. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003), no. 3, 971–1018.
  • [9] J. A. Carrillo, R. J. McCann and C. Villani. Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media. Archive for Rational Mechanics and Analysis, Volume 179, Issue 2, pp 217–263, 2006.
  • [10] P. Cattiaux, A. Guillin, and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1–2):19–40, 2008.
  • [11] D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31, no. 1, 29–85, 1983.
  • [12] M. H. Duong. Long time behaviour and particle approximation of a generalised Vlasov dynamic. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 127:1–16, 2015.
  • [13] M. H. Duong and J. Tugaut. Stationary solutions of the Vlasov–Fokker–Planck equation: Existence, characterization and phase-transition. Applied Mathematics Letters, 52:38–45, 2016.
  • [14] M. H. Duong, A. Lamacz, M. A. Peletier, and U. Sharma. Variational approach to coarse-graining of generalized gradient flows. Calculus of Variations and Partial Differential Equations, 56(4), 56:100, 2017.
  • [15] S. Herrmann and J. Tugaut. Non-uniqueness of stationary measures for self-stabilizing processes. Stochastic Process. Appl., 120(7):1215–1246, 2010.
  • [16] S. Herrmann and J. Tugaut: Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit. Electron. J. Probab., 15:2087–2116, 2010.
  • [17] S. Herrmann and J. Tugaut: Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit. ESAIM Probability and statistics, 2012.
  • [18] S. Herrmann, P. Imkeller, and D. Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008.
  • [19] H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4):284–304, 1940.
  • [20] T. Lelièvre, M. Rousset and G. Stoltz. Free energy computations: a mathematical perspective. Imperial College Press, 2010.
  • [21] F. Malrieu. Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab., 1(2):540–560, 2003.
  • [22] H. P. McKean, Jr. A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A., 56:1907–1911, 1966.
  • [23] H. P. McKean, Jr. Propagation of chaos for a class of nonlinear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41–57. Air Force Office Sci. Res., Arlington, Va., 1967.
  • [24] P. Monmarché. Long-time behaviour and propagation of chaos for mean field kinetic particles. Stochastic Processes and their Applications Volume 127, Issue 6, Pages 1721–1737, 2017.
  • [25] Y. Tamura. On asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31, no. 1, 195–221, 1984.
  • [26] Y. Tamura. Free energy and the convergence of distributions of diffusion processes of McKean type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, no. 2, 443–484, 1987.
  • [27] J. Tugaut. Convergence to the equilibria for self-stabilizing processes in double-well landscape. Ann. Probab. 41 (2013), no. 3A, 1427–1460
  • [28] J. Tugaut. Self-stabilizing processes in multi-wells landscape in $\mathbb{R} ^d$ - Convergence. Stochastic Processes and Their Applications http://dx.doi.org/10.1016/j.spa.2012.12.003, 2013.
  • [29] J. Tugaut. Self-stabilizing Processes in Multi-wells Landscape in $\mathbb{R} ^d$-Invariant Probabilities. Journal of Theoretical Probability, Volume 27, Issue 1, pp 57–79, 2014.
  • [30] J. Tugaut. Phase transitions of McKean-Vlasov processes in double-wells landscape. Stochastics 86 (2014), no. 2, 257–284